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The Normal (Gaussian) Distribution Section 4.3 1. Most used continuous distribution (as probability model) in statistics • Also known as Bell Curve 2. Is used for many physical measurements – heights, weights, test scores (also for errors in measurement) 3. “Central Limit Theorem” - sums, averages of random variables are often (close to) normally distributed 4. A two parameter “family” of distributions • 1 The Normal (Gaussian) Distribution Notation: ~ short hand for (is distributed as) Definition (via pdf): 2 1 The Normal (Gaussian) Distribution Geometry 3 The Normal (Gaussian) Distribution 4 2 The Standard Normal Distribution Special case: = 0 and 2 = 1 (Standardized Scores) cdf: Can not be expressed in closed functional form!` 5 Standard Normal Distribution CDF Table A3 provides (z) for z = -3.49, -3.48, ..., 3.48, 3.49 (-3.49) = 0.0002 and (3.49) = 1 - (-3.49) = 0.9998 6 3 Standard Normal Distribution CDF (Table Exercise) Example: Find P(Z ≤ 1.25) = Example: Find z such that P(Z ≤ z) = 0.05 (5th percentile) 7 8 4 9 Transforming (Standardizing) Normal RV’s Idea: Transform a N(, 2) RV into a N(0, 1) RV... then: 10 5 Using the Transformation Say and we want to compute Idea: Transform to the standard normal distribution: 11 Using the Transformation: Example Say the reaction time for a person to respond to brake lights is normally distributed, with a mean reaction time of 1.25 sec and a standard deviation of 0.46 sec. What is the probability that the reaction time for a randomly selected person is between 1.00 sec and 1.75 sec? 12 6 13 Using the Transformation: Percentiles Again say X N(1.25, 2 = (0.46)2 ). What is the 66th percentile of X? Compute 66th percentile of standard normal distribution and then transform to get 66th percentile for N(1.25, 2 = (0.46)2) distribution. 14 7 Using the Transformation: Percentiles Again say X N(1.25, 2 = (0.46)2 ). What is the 66th percentile of X? 15 8